Proving isomorphism between two sets

Question

Let A and B be two sets. Let : A→B be a one to one correspondence. Show that (P(A),⊊) and (P(B),⊊) are order isomorphic.

I am kinda lost with this question. How can I prove that these two are order isomorphic?

Answer 1

Hint: Every function $f: A \to B$ between two sets induces a natural function $f^*: P(A) \to P(B)$ given by $f^*(X) = \{ f(x) : x \in X \} \subseteq B$. This function $f^*$ clearly preserves inclusions.